Fit log returns to F-S skew standardized Student-t
distribution.
m is the location parameter.
s is the scale parameter.
nu is the estimated shape parameter (degrees of
freedom).
xi is the estimated skewness parameter.
For 2011, medium risk data is used in the high risk data set, as no
high risk fund data is available prior to 2012.
vmrl is a long version of Velliv medium risk data, from
2007 to 2023. For 2007 to 2011 (both included) no high risk data is
available.
The summary statistics are transformed back to the scale of gross returns by taking \(exp()\) of each summary statistic. (Note: Taking arithmetic mean of gross returns directly is no good. Must be geometric mean.)
| vmr | vhr | vmrl | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|---|
| Min. : | 0.868 | 0.849 | 0.801 | 0.904 | 0.878 | 0.988 | 0.977 | 0.979 | 0.967 |
| 1st Qu.: | 1.044 | 1.039 | 1.013 | 1.042 | 1.068 | 1.013 | 1.013 | 1.021 | 1.011 |
| Median : | 1.097 | 1.099 | 1.085 | 1.084 | 1.128 | 1.085 | 1.113 | 1.102 | 1.094 |
| Mean : | 1.067 | 1.080 | 1.057 | 1.063 | 1.089 | 1.064 | 1.085 | 1.079 | 1.072 |
| 3rd Qu.: | 1.136 | 1.160 | 1.128 | 1.107 | 1.182 | 1.101 | 1.128 | 1.121 | 1.107 |
| Max. : | 1.168 | 1.214 | 1.193 | 1.141 | 1.208 | 1.133 | 1.207 | 1.178 | 1.163 |
| Min. : | ranking | 1st Qu.: | ranking | Median : | ranking | Mean : | ranking | 3rd Qu.: | ranking | Max. : | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 0.988 | mmr | 1.068 | phr | 1.128 | phr | 1.089 | phr | 1.182 | phr | 1.214 | vhr |
| 0.979 | vmr_phr | 1.044 | vmr | 1.113 | mhr | 1.085 | mhr | 1.160 | vhr | 1.208 | phr |
| 0.977 | mhr | 1.042 | pmr | 1.102 | vmr_phr | 1.080 | vhr | 1.136 | vmr | 1.207 | mhr |
| 0.967 | vhr_pmr | 1.039 | vhr | 1.099 | vhr | 1.079 | vmr_phr | 1.128 | vmrl | 1.193 | vmrl |
| 0.904 | pmr | 1.021 | vmr_phr | 1.097 | vmr | 1.072 | vhr_pmr | 1.128 | mhr | 1.178 | vmr_phr |
| 0.878 | phr | 1.013 | vmrl | 1.094 | vhr_pmr | 1.067 | vmr | 1.121 | vmr_phr | 1.168 | vmr |
| 0.868 | vmr | 1.013 | mmr | 1.085 | vmrl | 1.064 | mmr | 1.107 | pmr | 1.163 | vhr_pmr |
| 0.849 | vhr | 1.013 | mhr | 1.085 | mmr | 1.063 | pmr | 1.107 | vhr_pmr | 1.141 | pmr |
| 0.801 | vmrl | 1.011 | vhr_pmr | 1.084 | pmr | 1.057 | vmrl | 1.101 | mmr | 1.133 | mmr |
Correlations
| vmr | vhr | pmr | phr | |
|---|---|---|---|---|
| vmr | 1.000 | 0.993 | -0.197 | -0.095 |
| vhr | 0.993 | 1.000 | -0.119 | -0.016 |
| pmr | -0.197 | -0.119 | 1.000 | 0.957 |
| phr | -0.095 | -0.016 | 0.957 | 1.000 |
Covariances
| vmr | vhr | pmr | phr | |
|---|---|---|---|---|
| vmr | 0.007 | 0.009 | -0.001 | -0.001 |
| vhr | 0.009 | 0.011 | -0.001 | 0.000 |
| pmr | -0.001 | -0.001 | 0.004 | 0.007 |
| phr | -0.001 | 0.000 | 0.007 | 0.011 |
Risk of loss at least as big as x percent for a single
period (year).
x values are row names.
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| 0 | 21.167 | 21.333 | 11.833 | 14.000 | 12.333 | 12.667 | 16.667 | 16.000 |
| 5 | 12.167 | 13.167 | 5.667 | 8.333 | 5.833 | 3.833 | 8.667 | 8.167 |
| 10 | 7.000 | 8.000 | 3.000 | 5.000 | 2.833 | 0.500 | 4.333 | 4.167 |
| 25 | 1.333 | 1.500 | 0.500 | 1.000 | 0.333 | 0.000 | 0.333 | 0.333 |
| 50 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 90 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 99 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 0 | ranking | 5 | ranking | 10 | ranking | 25 | ranking | 50 | ranking | 90 | ranking | 99 | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 21.333 | vhr | 13.167 | vhr | 8.000 | vhr | 1.500 | vhr | 0 | vmr | 0 | vmr | 0 | vmr |
| 21.167 | vmr | 12.167 | vmr | 7.000 | vmr | 1.333 | vmr | 0 | vhr | 0 | vhr | 0 | vhr |
| 16.667 | vmr_phr | 8.667 | vmr_phr | 5.000 | phr | 1.000 | phr | 0 | pmr | 0 | pmr | 0 | pmr |
| 16.000 | vhr_pmr | 8.333 | phr | 4.333 | vmr_phr | 0.500 | pmr | 0 | phr | 0 | phr | 0 | phr |
| 14.000 | phr | 8.167 | vhr_pmr | 4.167 | vhr_pmr | 0.333 | mmr | 0 | mmr | 0 | mmr | 0 | mmr |
| 12.667 | mhr | 5.833 | mmr | 3.000 | pmr | 0.333 | vmr_phr | 0 | mhr | 0 | mhr | 0 | mhr |
| 12.333 | mmr | 5.667 | pmr | 2.833 | mmr | 0.333 | vhr_pmr | 0 | vmr_phr | 0 | vmr_phr | 0 | vmr_phr |
| 11.833 | pmr | 3.833 | mhr | 0.500 | mhr | 0.000 | mhr | 0 | vhr_pmr | 0 | vhr_pmr | 0 | vhr_pmr |
Chance of gains of at least x percent for a single
period (year).
x values are row names.
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| 0 | 78.833 | 78.667 | 88.167 | 86.000 | 87.667 | 87.333 | 83.333 | 84.000 |
| 5 | 63.833 | 66.667 | 71.667 | 76.000 | 71.667 | 70.167 | 69.333 | 69.000 |
| 10 | 40.833 | 50.167 | 32.500 | 59.667 | 35.500 | 46.000 | 47.167 | 43.833 |
| 25 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.833 | 0.000 | 0.000 |
| 50 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 100 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 0 | ranking | 5 | ranking | 10 | ranking | 25 | ranking | 50 | ranking | 100 | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 88.167 | pmr | 76.000 | phr | 59.667 | phr | 0.833 | mhr | 0 | vmr | 0 | vmr |
| 87.667 | mmr | 71.667 | pmr | 50.167 | vhr | 0.000 | vmr | 0 | vhr | 0 | vhr |
| 87.333 | mhr | 71.667 | mmr | 47.167 | vmr_phr | 0.000 | vhr | 0 | pmr | 0 | pmr |
| 86.000 | phr | 70.167 | mhr | 46.000 | mhr | 0.000 | pmr | 0 | phr | 0 | phr |
| 84.000 | vhr_pmr | 69.333 | vmr_phr | 43.833 | vhr_pmr | 0.000 | phr | 0 | mmr | 0 | mmr |
| 83.333 | vmr_phr | 69.000 | vhr_pmr | 40.833 | vmr | 0.000 | mmr | 0 | mhr | 0 | mhr |
| 78.833 | vmr | 66.667 | vhr | 35.500 | mmr | 0.000 | vmr_phr | 0 | vmr_phr | 0 | vmr_phr |
| 78.667 | vhr | 63.833 | vmr | 32.500 | pmr | 0.000 | vhr_pmr | 0 | vhr_pmr | 0 | vhr_pmr |
Risk of loss at least as big as row name in percent from first to last period.
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| 0 | 4.81 | 2.84 | 1.74 | 1.07 | 0.41 | 0.10 | 0.16 | 0.18 |
| 5 | 4.16 | 2.39 | 1.59 | 1.02 | 0.33 | 0.08 | 0.15 | 0.16 |
| 10 | 3.49 | 1.93 | 1.36 | 0.93 | 0.28 | 0.06 | 0.12 | 0.11 |
| 25 | 2.12 | 1.06 | 1.01 | 0.60 | 0.10 | 0.03 | 0.04 | 0.08 |
| 50 | 0.87 | 0.33 | 0.51 | 0.31 | 0.02 | 0.01 | 0.01 | 0.01 |
| 90 | 0.07 | 0.03 | 0.09 | 0.03 | 0.00 | 0.00 | 0.00 | 0.00 |
| 99 | 0.02 | 0.01 | 0.05 | 0.01 | 0.00 | 0.00 | 0.00 | 0.00 |
1e6 simulation paths of mhr:
| 0 | 5 | 10 | 25 | 50 | 90 | 99 | |
|---|---|---|---|---|---|---|---|
| prob_pct | 0.118 | 0.095 | 0.076 | 0.036 | 0.008 | 0 | 0 |
| 0 | ranking | 5 | ranking | 10 | ranking | 25 | ranking | 50 | ranking | 90 | ranking | 99 | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 4.81 | vmr | 4.16 | vmr | 3.49 | vmr | 2.12 | vmr | 0.87 | vmr | 0.09 | pmr | 0.05 | pmr |
| 2.84 | vhr | 2.39 | vhr | 1.93 | vhr | 1.06 | vhr | 0.51 | pmr | 0.07 | vmr | 0.02 | vmr |
| 1.74 | pmr | 1.59 | pmr | 1.36 | pmr | 1.01 | pmr | 0.33 | vhr | 0.03 | vhr | 0.01 | vhr |
| 1.07 | phr | 1.02 | phr | 0.93 | phr | 0.60 | phr | 0.31 | phr | 0.03 | phr | 0.01 | phr |
| 0.41 | mmr | 0.33 | mmr | 0.28 | mmr | 0.10 | mmr | 0.02 | mmr | 0.00 | mmr | 0.00 | mmr |
| 0.18 | vhr_pmr | 0.16 | vhr_pmr | 0.12 | vmr_phr | 0.08 | vhr_pmr | 0.01 | mhr | 0.00 | mhr | 0.00 | mhr |
| 0.16 | vmr_phr | 0.15 | vmr_phr | 0.11 | vhr_pmr | 0.04 | vmr_phr | 0.01 | vmr_phr | 0.00 | vmr_phr | 0.00 | vmr_phr |
| 0.10 | mhr | 0.08 | mhr | 0.06 | mhr | 0.03 | mhr | 0.01 | vhr_pmr | 0.00 | vhr_pmr | 0.00 | vhr_pmr |
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| 0 | 95.19 | 97.16 | 98.26 | 98.93 | 99.59 | 99.90 | 99.84 | 99.82 |
| 5 | 94.52 | 96.84 | 98.03 | 98.84 | 99.49 | 99.89 | 99.77 | 99.78 |
| 10 | 93.73 | 96.43 | 97.84 | 98.68 | 99.37 | 99.88 | 99.75 | 99.74 |
| 25 | 91.25 | 94.88 | 97.12 | 98.20 | 98.85 | 99.79 | 99.50 | 99.53 |
| 50 | 85.80 | 91.67 | 95.25 | 97.31 | 97.56 | 99.47 | 98.99 | 98.90 |
| 100 | 72.20 | 83.01 | 88.55 | 94.67 | 89.93 | 97.65 | 96.24 | 94.23 |
| 200 | 39.42 | 60.89 | 59.51 | 85.27 | 48.53 | 86.42 | 80.34 | 66.41 |
| 300 | 16.21 | 39.24 | 22.32 | 70.63 | 11.13 | 62.93 | 52.20 | 30.58 |
| 400 | 5.17 | 23.88 | 4.42 | 54.36 | 1.26 | 37.79 | 25.70 | 9.69 |
| 500 | 1.51 | 12.81 | 0.54 | 38.37 | 0.09 | 18.77 | 9.84 | 2.54 |
| 1000 | 0.00 | 0.28 | 0.01 | 2.15 | 0.02 | 0.06 | 0.00 | 0.00 |
1e6 simulation paths of mhr:
| 0 | 5 | 10 | 25 | 50 | 100 | 200 | 300 | 400 | 500 | 1000 | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| prob | 99.882 | 99.854 | 99.824 | 99.686 | 99.301 | 97.513 | 86.912 | 65.992 | 41.486 | 21.693 | 0.086 |
| 0 | ranking | 5 | ranking | 10 | ranking | 25 | ranking | 50 | ranking | 100 | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 99.90 | mhr | 99.89 | mhr | 99.88 | mhr | 99.79 | mhr | 99.47 | mhr | 97.65 | mhr |
| 99.84 | vmr_phr | 99.78 | vhr_pmr | 99.75 | vmr_phr | 99.53 | vhr_pmr | 98.99 | vmr_phr | 96.24 | vmr_phr |
| 99.82 | vhr_pmr | 99.77 | vmr_phr | 99.74 | vhr_pmr | 99.50 | vmr_phr | 98.90 | vhr_pmr | 94.67 | phr |
| 99.59 | mmr | 99.49 | mmr | 99.37 | mmr | 98.85 | mmr | 97.56 | mmr | 94.23 | vhr_pmr |
| 98.93 | phr | 98.84 | phr | 98.68 | phr | 98.20 | phr | 97.31 | phr | 89.93 | mmr |
| 98.26 | pmr | 98.03 | pmr | 97.84 | pmr | 97.12 | pmr | 95.25 | pmr | 88.55 | pmr |
| 97.16 | vhr | 96.84 | vhr | 96.43 | vhr | 94.88 | vhr | 91.67 | vhr | 83.01 | vhr |
| 95.19 | vmr | 94.52 | vmr | 93.73 | vmr | 91.25 | vmr | 85.80 | vmr | 72.20 | vmr |
| 200 | ranking | 300 | ranking | 400 | ranking | 500 | ranking | 1000 | ranking |
|---|---|---|---|---|---|---|---|---|---|
| 86.42 | mhr | 70.63 | phr | 54.36 | phr | 38.37 | phr | 2.15 | phr |
| 85.27 | phr | 62.93 | mhr | 37.79 | mhr | 18.77 | mhr | 0.28 | vhr |
| 80.34 | vmr_phr | 52.20 | vmr_phr | 25.70 | vmr_phr | 12.81 | vhr | 0.06 | mhr |
| 66.41 | vhr_pmr | 39.24 | vhr | 23.88 | vhr | 9.84 | vmr_phr | 0.02 | mmr |
| 60.89 | vhr | 30.58 | vhr_pmr | 9.69 | vhr_pmr | 2.54 | vhr_pmr | 0.01 | pmr |
| 59.51 | pmr | 22.32 | pmr | 5.17 | vmr | 1.51 | vmr | 0.00 | vmr |
| 48.53 | mmr | 16.21 | vmr | 4.42 | pmr | 0.54 | pmr | 0.00 | vmr_phr |
| 39.42 | vmr | 11.13 | mmr | 1.26 | mmr | 0.09 | mmr | 0.00 | vhr_pmr |
Summary for fit of log returns to an F-S skew standardized Student-t
distribution.
m is the location parameter.
s is the scale parameter.
nu is the estimated degrees of freedom, or shape
parameter.
xi is the estimated skewness parameter.
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| m | 0.048 | 0.063 | 0.058 | 0.084 | 0.059 | 0.082 | 0.067 | 0.062 |
| s | 0.120 | 0.126 | 0.123 | 0.121 | 0.088 | 0.071 | 0.091 | 0.090 |
| nu | 3.304 | 4.390 | 2.265 | 3.185 | 2.773 | 89.863 | 4.660 | 3.892 |
| xi | 0.034 | 0.019 | 0.477 | 0.018 | 0.029 | 0.770 | 0.048 | 0.019 |
| R^2 | 0.993 | 0.995 | 0.991 | 0.964 | 0.890 | 0.961 | 0.927 | 0.933 |
| m | ranking | s | ranking | R^2 | ranking |
|---|---|---|---|---|---|
| 0.084 | phr | 0.071 | mhr | 0.995 | vhr |
| 0.082 | mhr | 0.088 | mmr | 0.993 | vmr |
| 0.067 | vmr_phr | 0.090 | vhr_pmr | 0.991 | pmr |
| 0.063 | vhr | 0.091 | vmr_phr | 0.964 | phr |
| 0.062 | vhr_pmr | 0.120 | vmr | 0.961 | mhr |
| 0.059 | mmr | 0.121 | phr | 0.933 | vhr_pmr |
| 0.058 | pmr | 0.123 | pmr | 0.927 | vmr_phr |
| 0.048 | vmr | 0.126 | vhr | 0.890 | mmr |
Monte Carlo simulations of portfolio index values (currency
values).
Statistics are given for the final state of all paths.
Probability of down-and_out is calculated as the share of paths that
reach 0 at some point. All subsequent values for a path are set to 0, if
the path reaches at any point.
0 is defined as any value below a threshold.
dai_pct (for down-and-in) is the probability of losing
money. This is calculated as the share of paths finishing below index
100.
## Number of paths: 10000
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| mc_m | 295.71 | 406.85 | 344.57 | 601.85 | 319.20 | 504.75 | 451.78 | 378.19 |
| mc_s | 135.11 | 211.26 | 114.07 | 273.01 | 102.36 | 173.81 | 152.58 | 121.44 |
| mc_min | 0.12 | 0.55 | 0.00 | 0.01 | 40.95 | 26.81 | 47.30 | 40.29 |
| mc_max | 1036.78 | 1504.07 | 1308.32 | 1930.64 | 4106.75 | 1414.95 | 1125.82 | 1097.82 |
| dao_pct | 0.00 | 0.00 | 0.03 | 0.01 | 0.00 | 0.00 | 0.00 | 0.00 |
| dai_pct | 4.47 | 2.46 | 1.66 | 0.99 | 0.42 | 0.10 | 0.14 | 0.15 |
| mc_m | ranking | mc_s | ranking | mc_min | ranking | mc_max | ranking | dao_pct | ranking | dai_pct | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 601.85 | phr | 102.36 | mmr | 47.30 | vmr_phr | 4106.75 | mmr | 0.00 | vmr | 0.10 | mhr |
| 504.75 | mhr | 114.07 | pmr | 40.95 | mmr | 1930.64 | phr | 0.00 | vhr | 0.14 | vmr_phr |
| 451.78 | vmr_phr | 121.44 | vhr_pmr | 40.29 | vhr_pmr | 1504.07 | vhr | 0.00 | mmr | 0.15 | vhr_pmr |
| 406.85 | vhr | 135.11 | vmr | 26.81 | mhr | 1414.95 | mhr | 0.00 | mhr | 0.42 | mmr |
| 378.19 | vhr_pmr | 152.58 | vmr_phr | 0.55 | vhr | 1308.32 | pmr | 0.00 | vmr_phr | 0.99 | phr |
| 344.57 | pmr | 173.81 | mhr | 0.12 | vmr | 1125.82 | vmr_phr | 0.00 | vhr_pmr | 1.66 | pmr |
| 319.20 | mmr | 211.26 | vhr | 0.01 | phr | 1097.82 | vhr_pmr | 0.01 | phr | 2.46 | vhr |
| 295.71 | vmr | 273.01 | phr | 0.00 | pmr | 1036.78 | vmr | 0.03 | pmr | 4.47 | vmr |
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| m | 0.064 | 0.077 | 0.061 | 0.085 | 0.062 | 0.081 | 0.076 | 0.069 |
| s | 0.081 | 0.099 | 0.063 | 0.101 | 0.048 | 0.070 | 0.062 | 0.060 |
Probability in percent that the smallest and largest (respectively) observed return for each fund was generated by a normal distribution:
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| P_norm(X_min) | 0.571 | 0.758 | 0.511 | 1.676 | 5.971 | 6.842 | 5.945 | 4.228 |
| P_norm(X_max) | 13.230 | 11.876 | 12.922 | 15.359 | 9.628 | 6.429 | 7.796 | 8.592 |
| P_t(X_min) | 5.377 | 5.080 | 3.489 | 4.315 | 10.570 | 8.015 | 13.008 | 10.520 |
| P_t(X_max) | 0.118 | 0.156 | 2.825 | 0.188 | 0.488 | 5.141 | 0.229 | 0.175 |
Average number of years between min or max events (respectively):
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| norm: avg yrs btw min | 175.248 | 131.911 | 195.568 | 59.669 | 16.748 | 14.616 | 16.820 | 23.650 |
| norm: avg yrs btw max | 7.559 | 8.420 | 7.739 | 6.511 | 10.386 | 15.556 | 12.827 | 11.639 |
| t: avg yrs btw min | 18.596 | 19.687 | 28.663 | 23.173 | 9.461 | 12.476 | 7.688 | 9.506 |
| t: avg yrs btw max | 848.548 | 640.410 | 35.400 | 531.552 | 205.104 | 19.450 | 437.280 | 572.483 |
p-values for Lilliefors test.
Testing \(H_0\), that log-returns are
Gaussian.
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| p value | 0.052 | 0.343 | 0.024 | 0.06 | 0.24 | 0.137 | 0.375 | 0.415 |
For different given probabilities that returns are Gaussian, what is the probability that the distribution is Gaussian rather than skewed t-distributed, given the smallest/largest observed log-returns?
Conditional probabilities for smallest observed log-returns:
Use \(1 - \text{p-value}\) from
Lilliefors test as prior probability that the distribution is
Gaussian.
\(x_{\text{obs}} = \min(x)\) and \(P[\text{Event}\ |\ \text{Gaussian}] =
P_{\text{Gauss}}[X \leq x_{\text{min}}]\):
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| Lillie p-val | 0.052 | 0.343 | 0.024 | 0.060 | 0.240 | 0.137 | 0.375 | 0.415 |
| Prior prob | 0.948 | 0.657 | 0.976 | 0.940 | 0.760 | 0.863 | 0.625 | 0.585 |
| P[Gauss | Event] | 0.661 | 0.088 | 0.960 | 0.754 | 0.839 | 0.917 | 0.653 | 0.603 |
Use \(1 - \text{p-value}\) from
Lilliefors test as prior probability that the distribution is
Gaussian.
\(x_{\text{obs}} = \max(x)\) and \(P[\text{Event}\ |\ \text{Gaussian}] =
P_{\text{Gauss}}[X \geq x_{\text{max}}]\):
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| Lillie p-val | 0.052 | 0.343 | 0.024 | 0.060 | 0.240 | 0.137 | 0.375 | 0.415 |
| Prior prob | 0.948 | 0.657 | 0.976 | 0.940 | 0.760 | 0.863 | 0.625 | 0.585 |
| P[Gauss | Event] | 1.000 | 0.986 | 0.997 | 0.998 | 0.993 | 0.888 | 0.988 | 0.991 |
Let’s plot the fit and the observed returns together.
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
Max vs sum plots for the first four moments:
Parameters
## [1] 1.2294983 0.3373312
Objective function plots
Let’s plot the fit and the observed returns together.
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
Max vs sum plots for the first four moments:
Parameters
## [1] 1.5074609 0.4255322
Objective function plots
Let’s plot the fit and the observed returns together.
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
Max vs sum plots for the first four moments:
Parameters
## [1] 1.2936284 0.3062685
Objective function plots
Let’s plot the fit and the observed returns together.
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
Max vs sum plots for the first four moments:
Parameters
## [1] 1.8379614 0.4397688
Objective function plots
Let’s plot the fit and the observed returns together.
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
Max vs sum plots for the first four moments:
Parameters
## [1] 1.1948623 0.2654885
Objective function plots
Let’s plot the fit and the observed returns together.
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
Max vs sum plots for the first four moments:
Parameters
## [1] 1.6413478 0.3380133
Objective function plots
Let’s plot the fit and the observed returns together.
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
Max vs sum plots for the first four moments:
Parameters
## [1] 1.5363616 0.3304634
Objective function plots
Let’s plot the fit and the observed returns together.
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
Max vs sum plots for the first four moments:
Parameters
## [1] 1.3625460 0.3050122
Objective function plots
num_paths = 1e61e6 paths:
Compare \(10^6\) and \(10^4\) paths for mhr:
| mc_m | mc_s | mc_min | mc_max | dao_pct | dai_pct | |
|---|---|---|---|---|---|---|
| mc_mhr_1e6 | 505.90695 | 173.22176 | 21.09569 | 1734.83520 | 0.00000 | 0.07330 |
| mc_mhr_1e4 | 504.75125 | 173.80504 | 26.81367 | 1414.94530 | 0.00000 | 0.10000 |
| is_mhr_1e4 | 510.836 | 2331.167 | 205.398 | 232384.846 | ibid. | ibid. |
Let \(m\) be the number of steps in each path and \(n\) be the number of paths. \(a\) is the initial capital. Use arithmetic mean for mean of all paths at time \(t\): \[\dfrac{a (e^{z_1} + e^{z_2} + \dots + e^{z_n})}{n}\] where \[z_i := x_{i, 1} + x_{i, 2} + \dots + x_{i, m}\] Use geometric mean for mean of all steps in a single path \(i\): \[a e^{\frac{x_{i, 1} + x_{i, 2} + \dots + x_{i, m}}{m}} = a \sqrt[m]{e^{x_{i, 1} + x_{i, 2} + \dots + x_{i, m}}}\]
So for Monte Carlo of returns after \(m\) periods, we
For Importance Sampling, we
\[\text{Avg. of returns} := \dfrac{ \left(\dfrac{x_t}{x_{t-1}} + \dfrac{y_t}{y_{t-1}} \right) }{2}\] \[\text{Returns of avg.} := \left(\dfrac{ x_t + y_t }{2}\right) \Big/ \left(\dfrac{ x_{t-1} + y_{t-1} }{2}\right) \equiv \dfrac{ x_t + y_t }{ x_{t-1} + y_{t-1}}\]
For which \(x_1\) and \(y_1\) are \(\text{Avg. of returns} = \text{Returns of avg.}\)?
\[\dfrac{ \left(\dfrac{x_t}{x_{t-1}} + \dfrac{y_t}{y_{t-1}} \right) }{2} = \dfrac{ x_t + y_t }{ x_{t-1} + y_{t-1}}\]
\[\dfrac{x_t}{x_{t-1}} + \dfrac{y_t}{y_{t-1}} = 2 \dfrac{ x_t + y_t }{ x_{t-1} + y_{t-1}}\]
\[(x_{t-1} + y_{t-1}) x_t y_{t-1} + (x_{t-1} + y_{t-1}) x_{t-1} y_t = 2 (x_{t-1}y_{t-1}x_t + x_{t-1}y_{t-1}y_t)\]
\[(x_{t-1}x_1y_{t-1} + y_{t-1}x_ty_{t-1}) + (x_{t-1}x_{t-1}y_t + x_{t-1}y_{t-1}y_t) = 2(x_{t-1}y_{t-1}x_t + x_{t-1}y_{t-1}y_t)\] This is not generally true, but true if for instance \(x_{t-1} = y_{t-1}\).
Definition: R = 1+r
## Let x_0 be 100.
## Let y_0 be 200.
## So the initial value of the pf is 300 .
## Let R_x be 0.5.
## Let R_y be 1.5.
Then,
## x_1 is R_x * x_0 = 50.
## y_1 is R_y * y_0 = 300.
Average of returns:
## 0.5 * (R_x + R_y) = 1
So here the value of the pf at t=1 should be unchanged from t=0:
## (x_0 + y_0) * 0.5 * (R_x + R_y) = 300
But this is clearly not the case:
## 0.5 * (x_1 + y_1) = 0.5 * (R_x * x_0 + R_y * y_0) = 175
Therefore we should take returns of average, not average of returns!
Let’s take the average of log returns instead:
## 0.5 * (log(R_x) + log(R_y)) = -0.143841
We now get:
## (x_0 + y_0) * exp(0.5 * (log(Rx) + log(Ry))) = 259.8076
So taking the average of log returns doesn’t work either.
Test if a simulation of a mix (average) of two returns series has the same distribution as a mix of two simulated returns series.
## m(data_x): 0.08883305
## s(data_x): 0.4589474
## m(data_y): 9.292777
## s(data_y): 2.750134
##
## m(data_x + data_y): 4.690805
## s(data_x + data_y): 1.346744
m and s of final state of all paths.
_a is mix of simulated returns.
_b is simulated mixed returns.
| m_a | m_b | s_a | s_b |
|---|---|---|---|
| 94.046 | 93.891 | 6.264 | 6.020 |
| 93.913 | 93.587 | 6.106 | 6.177 |
| 93.660 | 93.652 | 6.204 | 5.915 |
| 94.064 | 93.675 | 6.300 | 5.859 |
| 93.895 | 93.743 | 6.284 | 6.008 |
| 93.750 | 93.820 | 6.172 | 6.190 |
| 94.229 | 93.836 | 6.275 | 6.015 |
| 93.410 | 93.649 | 6.378 | 6.194 |
| 93.998 | 93.960 | 6.205 | 5.944 |
| 93.712 | 93.724 | 6.426 | 6.167 |
## m_a m_b s_a s_b
## Min. :93.41 Min. :93.59 Min. :6.106 Min. :5.859
## 1st Qu.:93.72 1st Qu.:93.66 1st Qu.:6.205 1st Qu.:5.960
## Median :93.90 Median :93.73 Median :6.269 Median :6.018
## Mean :93.87 Mean :93.75 Mean :6.261 Mean :6.049
## 3rd Qu.:94.03 3rd Qu.:93.83 3rd Qu.:6.296 3rd Qu.:6.174
## Max. :94.23 Max. :93.96 Max. :6.426 Max. :6.194
_a and _b are very close to equal.
We attribute the differences to differences in estimating the
distributions in version a and b.
The final state is independent of the order of the preceding steps:
So does the order of the steps in the two processes matter, when mixing simulated returns?
The order of steps in the individual paths do not matter, because the mix of simulated paths is a sum of a sum, so the order of terms doesn’t affect the sum. If there is variation it is because the sets preceding steps are not the same. For instance, the steps between step 1 and 60 in the plot above are not the same for the two lines.
Recall, \[\text{Var}(aX+bY) = a^2 \text{Var}(X) + b^2 \text{Var}(Y) + 2ab \text{Cov}(a, b)\]
var(0.5 * vhr + 0.5 * phr)
## [1] 0.005355618
0.5^2 * var(vhr) + 0.5^2 * var(phr) + 2 * 0.5 * 0.5 * cov(vhr, phr)
## [1] 0.005355618
Our distribution estimate is based on 13 observations. Is that enough
for a robust estimate? What if we suddenly hit a year like 2008? How
would that affect our estimate?
Let’s try to include the Velliv data from 2007-2010.
We do this by sampling 13 observations from vmrl.
## m s
## Min. :0.06156 Min. :0.04718
## 1st Qu.:0.06885 1st Qu.:0.06157
## Median :0.07107 Median :0.07015
## Mean :0.07218 Mean :0.07064
## 3rd Qu.:0.07693 3rd Qu.:0.08091
## Max. :0.08582 Max. :0.09479
xiThe fit for mhr has the highest xi value of
all. This suggests right-skew:
If the Law Of Large Numbers holds true, \[\dfrac{\max (X_1^p, ..., X^p)}{\sum_{i=1}^n X_i^p} \rightarrow 0\] for \(n \rightarrow \infty\).
If not, \(X\) doesn’t have a \(p\)’th moment.
See Taleb: The Statistical Consequences Of Fat Tails, p. 192
Comments
(Ignoring
mhr_a…)mhrhas some nice properties:nuvalue of 90, which means it is tending more towards exponential tails than polynomial tails. All other funds havenuvalues close to 3, exceptphrwhich is even worse at close to 2. (Note that for a Gaussian,nuis infinite.)phr.mmr, and less thanphr.phrhas a highermc_m.mc_sthan the individual components,vhrandphr.xiof all fits, suggesting less left skewness. Density plots forvmr,phrandmmrhave an extremely sharp drop, as if an upward limiter has been applied, which corresponds to extremely lowxivalues. The density plot formhris by far the most symmetrical of all the fits. As seen in the section “Compare Gaussian and skewed t-distribution fits”, the other skewed t-distribution fits don’t capture the max observed returns at all.mmrhas as highermc_min. However, that ofmmris 18 times higher with 62, sommris a clear winner here.mc_maxsmaller than the individual components,vhrandphr, but ca. 1.5 times higher thenmmr.Taleb, Statistical Consequences Of Fat Tails, p. 97:
“the variance of a finite variance random variable with tail exponent \(< 4\) will be infinite”.
And p. 363:
“The hedging errors for an option portfolio (under a daily revision regime) over 3000 days, un- der a constant volatility Student T with tail exponent \(\alpha = 3\). Technically the errors should not converge in finite time as their distribution has infinite variance.”